This article is from the Computer Viruses FAQ, by Nick FitzGerald n.fitzgerald@csc.canterbury.ac.nz with numerous contributions by others.
In general terms, the answer is no. E-mail messages and postings on
BBSes and News are text data and will not be executed as programs.
Computer viruses are programs, and must be executed to do anything, so
the simple act of reading online messages doesn't pose a threat of
catching a computer virus.
There are a few provisos to be made. If your computer uses ANSI screen
and keyboard controls, you may be susceptible to an ANSI bomb (see B14).
An ANSI bomb may, merely by being placed in text read on the screen,
temporarily redefine keys on the keyboard to perform various functions.
It is, however, very unlikely that you will ever see an ANSI bomb in
e-mail, or that it could do significant damage while you are reading
mail.
Another possibility is that mail can be used to send programs. To do
this program files have to be encoded into a special form so the binary
(8-bit) program files are not corrupted by transfer over the text-only
(7-bit) e-mail transport medium. Probably the commonest of these
encoding schemes is uuencoding, though there are several others. If you
receive an encoded program, you normally have to use a decoding program
or special option in your e-mail program to extract it and decode it
before it can be run. Once you have extracted the program though, you
should then treat it as you would any other program whose source you do
not know, and test it before you run it.
A third possibility is with the newer, highly-automated online systems.
Some of these attempt to make online access much easier for the user by
automating such features as file transfer and program updates. At least
one commercial online service is known to have the capability of sending
new programs to the user and to invoke those programs while the user is
still online. While there is no reason to assume that any service that
does this *will* infect you, any time things are going on that you are
not being told about, you are at greater risk.
 
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